Harmonic maximum principle on a finite graph
= Harmonic maximum principle on a finite graph
A real function $h$ on a finite <connected graph> satisfying $Lh=0$ everywhere is constant. At a maximum <graph vertex>, the identity $\deg(u)h(u)=\sum_{v\sim u}h(v)$ forces every <graph neighbour> to have the same maximum. Connectedness propagates it to every <graph vertex>. This proves uniqueness of <voltage> up to an additive constant, and identifies sums of oppositely directed occupation <voltages>.